The Theory of the Riemann Zeta-function


Author: Edward Charles Titchmarsh,D. R. Heath-Brown
Publisher: Oxford University Press
ISBN: 9780198533696
Category: Architecture
Page: 412
View: 8713

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The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects of the theory, starting from first principles and probing the function's own challenging theory, with the famous and still unsolved "Riemann hypothesis" at its heart. The second edition has been revised to include descriptions of work done in the last forty years and is updated with many additional references; it will provide stimulating reading for postgraduates and workers in analytic number theory and classical analysis.

The Theory of Functions


Author: Edward Charles Titchmarsh
Publisher: Oxford University Press
ISBN: 0198533497
Category: Mathematics
Page: 454
View: 9195

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The Theory of Functions

The Riemann Zeta-Function

Theory and Applications
Author: Aleksandar Ivic
Publisher: Courier Corporation
ISBN: 0486140040
Category: Mathematics
Page: 562
View: 6786

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This text covers exponential integrals and sums, 4th power moment, zero-free region, mean value estimates over short intervals, higher power moments, omega results, zeros on the critical line, zero-density estimates, and more. 1985 edition.

In Search of the Riemann Zeros

Strings, Fractal Membranes and Noncommutative Spacetimes
Author: Michel Laurent Lapidus
Publisher: American Mathematical Soc.
ISBN: 9780821842225
Category: Mathematics
Page: 558
View: 4446

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Formulated in 1859, the Riemann Hypothesis is the most celebrated and multifaceted open problem in mathematics. In essence, it states that the primes are distributed as harmoniously as possible--or, equivalently, that the Riemann zeros are located on a single vertical line, called the critical line. In this book, the author proposes a new approach to understand and possibly solve the Riemann Hypothesis. His reformulation builds upon earlier (joint) work on complex fractal dimensions and the vibrations of fractal strings, combined with string theory and noncommutative geometry. Accordingly, it relies on the new notion of a fractal membrane or quantized fractal string, along with the modular flow on the associated moduli space of fractal membranes. Conjecturally, under the action of the modular flow, the spacetime geometries become increasingly symmetric and crystal-like, hence, arithmetic. Correspondingly, the zeros of the associated zeta functions eventually condense onto the critical line, towards which they are attracted, thereby explaining why the Riemann Hypothesis must be true. Written with a diverse audience in mind, this unique book is suitable for graduate students, experts and nonexperts alike, with an interest in number theory, analysis, dynamical systems, arithmetic, fractal or noncommutative geometry, and mathematical or theoretical physics.

The Riemann Hypothesis and the Roots of the Riemann Zeta Function


Author: Samuel W. Gilbert
Publisher: Riemann hypothesis
ISBN: 9781439216385
Category: Mathematics
Page: 140
View: 1768

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The author demonstrates that the Dirichlet series representation of the Riemann zeta function converges geometrically at the roots in the critical strip. The Dirichlet series parts of the Riemann zeta function diverge everywhere in the critical strip. It has therefore been assumed for at least 150 years that the Dirichlet series representation of the zeta function is useless for characterization of the non-trivial roots. The author shows that this assumption is completely wrong. Reduced, or simplified, asymptotic expansions for the terms of the zeta function series parts are equated algebraically with reduced asymptotic expansions for the terms of the zeta function series parts with reflected argument, constraining the real parts of the roots of both functions to the critical line. Hence, the Riemann hypothesis is correct. Formulae are derived and solved numerically, yielding highly accurate values of the imaginary parts of the roots of the zeta function.

The Distribution of Prime Numbers


Author: Albert Edward Ingham
Publisher: Cambridge University Press
ISBN: 9780521397896
Category: Mathematics
Page: 114
View: 5854

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Originally published in 1934, this volume presents the theory of the distribution of the prime numbers in the series of natural numbers. Despite being long out of print, it remains unsurpassed as an introduction to the field.

An Introduction to the Theory of the Riemann Zeta-Function


Author: S. J. Patterson
Publisher: Cambridge University Press
ISBN: 131658335X
Category: Mathematics
Page: N.A
View: 974

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This is a modern introduction to the analytic techniques used in the investigation of zeta functions, through the example of the Riemann zeta function. Riemann introduced this function in connection with his study of prime numbers and from this has developed the subject of analytic number theory. Since then many other classes of 'zeta function' have been introduced and they are now some of the most intensively studied objects in number theory. Professor Patterson has emphasised central ideas of broad application, avoiding technical results and the customary function-theoretic approach. Thus, graduate students and non-specialists will find this an up-to-date and accessible introduction, especially for the purposes of algebraic number theory. There are many exercises included throughout, designed to encourage active learning.

Sieve Methods


Author: Heine Halberstam,Hans Egon Richert
Publisher: Courier Corporation
ISBN: 0486320804
Category: Mathematics
Page: 384
View: 6196

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This text by a noted pair of experts is regarded as the definitive work on sieve methods. It formulates the general sieve problem, explores the theoretical background, and illustrates significant applications. 1974 edition.

Exploring the Riemann Zeta Function

190 years from Riemann's Birth
Author: Hugh Montgomery,Ashkan Nikeghbali,Michael Th. Rassias
Publisher: Springer
ISBN: 3319599690
Category: Mathematics
Page: 298
View: 6190

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Exploring the Riemann Zeta Function: 190 years from Riemann's Birth presents a collection of chapters contributed by eminent experts devoted to the Riemann Zeta Function, its generalizations, and their various applications to several scientific disciplines, including Analytic Number Theory, Harmonic Analysis, Complex Analysis, Probability Theory, and related subjects. The book focuses on both old and new results towards the solution of long-standing problems as well as it features some key historical remarks. The purpose of this volume is to present in a unified way broad and deep areas of research in a self-contained manner. It will be particularly useful for graduate courses and seminars as well as it will make an excellent reference tool for graduate students and researchers in Mathematics, Mathematical Physics, Engineering and Cryptography.

The Theory of Hardy's Z-Function


Author: A. Ivić
Publisher: Cambridge University Press
ISBN: 1107028833
Category: Mathematics
Page: 245
View: 1689

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"This book is an outgrowth of a mini-course held at the Arctic Number Theory School, University of Helsinki, May 18-25, 2011. The central topic is Hardy's function, of great importance in the theory of the Riemann zeta-function. It is named after GodfreyHarold Hardy FRS (1877-1947), who was a prominent English mathematician, well-known for his achievements in number theory and mathematical analysis"--

Lectures on the Riemann Zeta Function


Author: H. Iwaniec
Publisher: American Mathematical Society
ISBN: 1470418517
Category: Mathematics
Page: 119
View: 7937

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The Riemann zeta function was introduced by L. Euler (1737) in connection with questions about the distribution of prime numbers. Later, B. Riemann (1859) derived deeper results about the prime numbers by considering the zeta function in the complex variable. The famous Riemann Hypothesis, asserting that all of the non-trivial zeros of zeta are on a critical line in the complex plane, is one of the most important unsolved problems in modern mathematics. The present book consists of two parts. The first part covers classical material about the zeros of the Riemann zeta function with applications to the distribution of prime numbers, including those made by Riemann himself, F. Carlson, and Hardy-Littlewood. The second part gives a complete presentation of Levinson's method for zeros on the critical line, which allows one to prove, in particular, that more than one-third of non-trivial zeros of zeta are on the critical line. This approach and some results concerning integrals of Dirichlet polynomials are new. There are also technical lemmas which can be useful in a broader context.

Exploring the Riemann Zeta Function

190 years from Riemann's Birth
Author: Hugh Montgomery,Ashkan Nikeghbali,Michael Th. Rassias
Publisher: Springer
ISBN: 3319599690
Category: Mathematics
Page: 298
View: 6073

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Exploring the Riemann Zeta Function: 190 years from Riemann's Birth presents a collection of chapters contributed by eminent experts devoted to the Riemann Zeta Function, its generalizations, and their various applications to several scientific disciplines, including Analytic Number Theory, Harmonic Analysis, Complex Analysis, Probability Theory, and related subjects. The book focuses on both old and new results towards the solution of long-standing problems as well as it features some key historical remarks. The purpose of this volume is to present in a unified way broad and deep areas of research in a self-contained manner. It will be particularly useful for graduate courses and seminars as well as it will make an excellent reference tool for graduate students and researchers in Mathematics, Mathematical Physics, Engineering and Cryptography.

The Mysteries of the Real Prime


Author: M. J. Shai Haran
Publisher: Oxford University Press
ISBN: 9780198508687
Category: Mathematics
Page: 240
View: 7936

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Highly topical and original monograph, introducing the author's work on the Riemann zeta function and its adelic interpretation of interest to a wide range of mathematicians and physicists.

The Riemann Hypothesis

A Resource for the Afficionado and Virtuoso Alike
Author: Peter Borwein
Publisher: Springer Science & Business Media
ISBN: 0387721258
Category: Mathematics
Page: 533
View: 8946

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This book presents the Riemann Hypothesis, connected problems, and a taste of the body of theory developed towards its solution. It is targeted at the educated non-expert. Almost all the material is accessible to any senior mathematics student, and much is accessible to anyone with some university mathematics. The appendices include a selection of original papers. This collection is not very large and encompasses only the most important milestones in the evolution of theory connected to the Riemann Hypothesis. The appendices also include some authoritative expository papers. These are the "expert witnesses” whose insight into this field is both invaluable and irreplaceable.

Riemann's Zeta Function


Author: Harold M. Edwards
Publisher: Courier Corporation
ISBN: 9780486417400
Category: Mathematics
Page: 315
View: 8692

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Superb high-level study of one of the most influential classics in mathematics examines landmark 1859 publication entitled “On the Number of Primes Less Than a Given Magnitude,” and traces developments in theory inspired by it. Topics include Riemann's main formula, the prime number theorem, the Riemann-Siegel formula, large-scale computations, Fourier analysis, and other related topics. English translation of Riemann's original document appears in the Appendix.

An Introduction to the Theory of Numbers


Author: Godfrey Harold Hardy,E. M. Wright,Roger Heath-Brown,Joseph Silverman
Publisher: Oxford University Press
ISBN: 9780199219865
Category: Mathematics
Page: 621
View: 6438

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An Introduction to the Theory of Numbers by G.H. Hardy and E. M. Wright is found on the reading list of virtually all elementary number theory courses and is widely regarded as the primary and classic text in elementary number theory. This Sixth Edition has been extensively revised and updated to guide today's students through the key milestones and developments in number theory. Updates include a chapter on one of the mostimportant developments in number theory -- modular elliptic curves and their role in the proof of Fermat's Last Theorem -- a foreword by A. Wiles and comprehensively updated end-of-chapter notes detailing the key developments in number theory. Suggestions for further reading are also included for the more avid reader and the clarityof exposition is retained throughout making this textbook highly accessible to undergraduates in mathematics from the first year upwards.

Prime Numbers and the Riemann Hypothesis


Author: Barry Mazur,William Stein
Publisher: Cambridge University Press
ISBN: 1107101921
Category: Mathematics
Page: 150
View: 4523

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This book introduces prime numbers and explains the famous unsolved Riemann hypothesis.

Multiplicative Number Theory


Author: H. Davenport
Publisher: Springer Science & Business Media
ISBN: 1475759274
Category: Mathematics
Page: 177
View: 6661

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Although it was in print for a short time only, the original edition of Multiplicative Number Theory had a major impact on research and on young mathematicians. By giving a connected account of the large sieve and Bombieri's theorem, Professor Davenport made accessible an important body of new discoveries. With this stimula tion, such great progress was made that our current understanding of these topics extends well beyond what was known in 1966. As the main results can now be proved much more easily. I made the radical decision to rewrite §§23-29 completely for the second edition. In making these alterations I have tried to preserve the tone and spirit of the original. Rather than derive Bombieri's theorem from a zero density estimate tor L timctions, as Davenport did, I have chosen to present Vaughan'S elementary proof of Bombieri's theorem. This approach depends on Vaughan's simplified version of Vinogradov's method for estimating sums over prime numbers (see §24). Vinogradov devised his method in order to estimate the sum LPH e(prx); to maintain the historical perspective I have inserted (in §§25, 26) a discussion of this exponential sum and its application to sums of primes, before turning to the large sieve and Bombieri's theorem. Before Professor Davenport's untimely death in 1969, several mathematicians had suggested small improvements which might be made in Multiplicative Number Theory, should it ever be reprinted.